
The \(n\) th Bell number represents the number of ways to group \(n\)  elements. The (\(n\) , \(k\) )th Stirling number of the second kind represents the number of ways to group \(n\)  elements by \(k\)  disjoint groups. Then, sum of the (\(n\) , \(k\) )th Stirling number of the second kind from \(k=0\)  to \(n\)  equals to the \(n\) th Bell number. While the (\(n\) , \(k\) )th unsigned Stirling numbers of the first kind can also be defined as coefficients in a linear combination of \(x^k\)  for \(k=0, \cdots , n\)  that equals to the rising factorial, \(x(x+1)\cdots (x+n-1)\) , the (\(n\) , \(k\) )th Stirling number of the second kind can also be defined as coefficients in a linear combination of the falling factorial \(x(x-1)\cdots (x-k+1)\)  for \(k=0, \cdots , n\)  that equals to \(x^n\)  [1]} (Fig. REF (b)). Furthermore, both the factorials and the Bell numbers have the modular property concerning prime numbers. The following theorem is called Wilson's theorem [2]}, [3]}, [4]} and states that the \((p-1)\) th factorial equals to \(-1\)  modulo \(p\)  if and only if \(p\)  is a prime number.
